9 Infinite Series Copyright Cengage Learning All rights

9 Infinite Series Copyright © Cengage Learning. All rights reserved.

9. 2 Series and Convergence Copyright © Cengage Learning. All rights reserved.

Objectives n Understand the definition of a convergent infinite series. n Use properties of infinite geometric series. n Use the nth-Term Test for Divergence of an infinite series. 3

Infinite Series 4

Infinite Series One important application of infinite sequences is in representing “infinite summations. ” Informally, if {an} is an infinite sequence, then is an infinite series (or simply a series). The numbers a 1, a 2, a 3, are the terms of the series. For some series it is convenient to begin the index at n = 0 (or some other integer). As a typesetting convention, it is common to represent an infinite series as simply 5

Infinite Series In such cases, the starting value for the index must be taken from the context of the statement. To find the sum of an infinite series, consider the following sequence of partial sums. If this sequence of partial sums converges, the series is said to converge. 6

Infinite Series 7

Example 1(a) – Convergent and Divergent Series The series has the following partial sums. 8

Example 1(a) – Convergent and Divergent Series cont’d Because it follows that the series converges and its sum is 1. 9

Example 1(b) – Convergent and Divergent Series cont’d The nth partial sum of the series is given by Because the limit of Sn is 1, the series converges and its sum is 1. 10

Example 1(c) – Convergent and Divergent Seriescont’d The series diverges because Sn = n and the sequence of partial sums diverges. 11

Infinite Series The series is a telescoping series of the form Note that b 2 is canceled by the second term, b 3 is canceled by the third term, and so on. 12

Infinite Series Because the nth partial sum of this series is S n = b 1 – bn + 1 it follows that a telescoping series will converge if and only if bn approaches a finite number as Moreover, if the series converges, its sum is 13

Geometric Series 14

Geometric Series The series. is a geometric In general, the series given by is a geometric series with ratio r. 15

Geometric Series 16

Example 3(a) – Convergent and Divergent Geometric Series The geometric series has a ratio of with a = 3. Because 0 < |r| < 1, the series converges and its sum is 17

Example 3(b) – Convergent and Divergent Geometric Series cont’d The geometric series has a ratio of Because |r| ≥ 1, the series diverges. 18

Geometric Series 19

nth-Term Test for Divergence 20

nth-Term Test for Divergence The contrapositive of Theorem 9. 8 provides a useful test for divergence. This nth-Term Test for Divergence states that if the limit of the nth term of a series does not converge to 0, the series must diverge. 21

Example 5 – Using the nth-Term Test for Divergence a. For the series you have So, the limit of the nth term is not 0, and the series diverges. b. For the series you have So, the limit of the nth term is not 0, and the series diverges. 22

Example 5 – Using the nth-Term Test for Divergence c. For the series cont’d you have Because the limit of the nth term is 0, the nth-Term Test for Divergence does not apply and you can draw no conclusions about convergence or divergence. 23